Summary: On the Hurewicz map and Postnikov invariants
of KZ
Christian Ausoni
Abstract. The purpose of this note is to present a calculation of the Hurewicz
homomorphism h : KZ -- H(GL(Z); Z) on the elements of KZ known
to generate direct summands. These results are then used to produce lower
bounds for the Postnikov invariants of the space KZ. Under extra hypothesises
(compatible with the Quillen-Lichtenbaum conjecture for Z), we give the exact
p-primary part of the order of the latter invariants.
1. Introduction
D. Quillen defined, for any integer n 1, the higher algebraic K-theory group
KnR of a ring R as the homotopy group KnR = n(BGL(R)+
). In this paper, we
will calculate the Hurewicz homomorphism
h : KZ - H(BGL(Z)+
; Z) = H(GL(Z); Z)
on elements of KZ that are known to generate direct summands. One motivation
for such a calculation is to obtain information on the homotopy type of the space
BGL(Z)+
, which we will denote in the sequel by KZ. Its weak homotopy type is